A comment is in order about the crossing of momenta in the presence of the parity-odd variable \(\varepsilon _5\)  and the parity-odd transcendental functions appearing in the basis of ref. [1]}. When the analytic expression for a given helicity is being derived, one needs to match scalar integrals with permuted momenta to the basis \(t_e\) . In practice, these scalar integrals are the master integrals resulting from the IBP reduction of the amplitude.
This matching is done in the following way. Ref. [1]} provides the transcendental basis \(t_e\)  indirectly, through a set of master integrals of uniform transcendentality (UTM). These UTMs depend on the invariants \(s_{ij}\)  but not on \(\varepsilon _5\) . Ref. [1]} also provides for a single permutation of momenta a way of expressing a set of scalar integrals to the set of UTMs. The parity-odd variable \(\varepsilon _5\)  appears linearly in this relation. This relation allows us to map our own scalar integrals to UTMs and, from there, to \(t_e\) .
